FACTORING, Adelic Complexity, and the Silent-Radix Principle

In the paper's words

The silent radix -- the observation that no positional numeral internally specifies its own base -- was previously explored as a cryptographic primitive through Silent Radix Encryption (SRE), a key encapsulation mechanism whose security rests on the hardness of recovering a secret base from public envelope digits. That scheme, while closing the known-plaintext attack vector via digit randomization, remains structurally linear: without noise, small numbers of ciphertexts leak algebraic constraints that lattice reduction can exploit. We present Noise-Augmented Silent Radix Encryption (N-SRE), which injects small p-adic errors into the envelope digits and formalizes the underlying hard problem as Hidden-Modulus Learning With Errors (HM-LWE).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Experiments E1-E6 of hm-lwe-experiments.py confirm: Sample generation correctness for $p$ up to 32 bits Theorem 1: known-$p$ Gaussian elimination recovers $s$ exactly $\forall t \in \{0,1,2,3,4\}$, with $O(n^3)$ scaling confirmed from $n=4$ (0.04 ms) to $n=128$ (135.7 ms) Statistical indistinguishability: $\chi_{100}^2 = 81.0$ (not significant) and byte-level Shannon entropy within 0.1% of uniform Lattice Attack Defeat.

   from 5.5. Experimental Verification

2. 14 KB 1,568 bytes 128 bytes Security assumption HM-LWE (unstandardized) MLWE (standardized) Goppa decoding Provable security Partial (known-$p$ reduction) Full (worst-case to average-case) Full Cryptographic capabilities KEM only KEM + signatures KEM only Maturity Research prototype NIST standard NIST standard N-SRE's ciphertext expansion (approximately 14 KB vs.

   from 6.2. Comparison with Standardized PQC

3. Experimental verification: We provide a complete computational verification suite (2,200+ lines of cryptanalysis code across four experiment notebooks) that validates every core claim: Theorem 1 (noise-cancellation), lattice-attack defeat, statistical indistinguishability, $O(n^3)$ scaling, and N-SRE round-trip correctness at 100% success rate across 500 random messages.

   from 1.3. Contributions

4. Setup($\lambda$): Choose a secret prime $p$ of bit-length $2\lambda$ (256 or 512 bits), digit length $k$, error parameters $t$ and $E$.

   from 4.3. Algorithm Specification

5. The n-sre-reference-impl.py notebook implements N-SRE-LWE with: Known-Answer Test vector verified (deterministic) 500/500 random messages (16-256 bytes) correctly encrypted/decrypted — 100% success rate at 0.26 ms/trial average Performance: 32-bit $p$, $n=8$: 0.09 ms encapsulate, 0.17 ms decapsulate; 128-bit $p$, $n=32$: 1.08 ms encapsulate, 7.85 ms decapsulate Attack Resistance.

   from 5.5. Experimental Verification

6. 13.6 KB Error valuation $t$ 4 4 Ciphertext (total) approx.

   from 6.1. Recommended Parameters

How it is done

1. We prove that HM-LWE reduces to standard LWE when the modulus $p$ is known to the adversary, and we argue that the unknown-modulus variant resists lattice reduction because the adversary lacks the correct geometry in which to construct a short vector.

   from Abstract

2. The hidden-modulus approach differs qualitatively: the adversary does not know the modulus $p$.

   from 1.2. Motivation: Why Hidden Modulus?

3. This paper makes the following contributions: HM-LWE problem formalization: We define the Hidden-Modulus Learning With Errors problem, specify its distribution, and establish its relationship to standard LWE (reduction when $p$ is known) and to the structured integer recovery problem (when noise is zero).

   from 1.3. Contributions

4. The encapsulation procedure is: Alice generates $k$ random decimal digits $D = [d_0, \ldots, d_{k-1}]$, $d_i \in \{0, \ldots, 9\}$.

   from 2.3. Silent Radix Encryption (SRE) -- Review

What would falsify it

1. If the decision-HM-LWE problem is hard, then the N-SRE-LWE variant is IND-CPA secure.

   from 4.5. Formal Security Model

Where the claim stops

1. While N-SRE does not yet match the bandwidth efficiency of NIST-standardized lattice KEMs, it demonstrates that the valuation-theoretic design space -- hiding not just a parameter but the mathematical topology of interpretation -- can support noise-augmented constructions with plausible hardness foundations.

   from Abstract

2. Security cannot be obtained from key entropy alone when the cipher is linear and deterministic.

   from 1.1 The Silent Radix and its Cryptographic Implications

3. The paper identified noise-augmented $p$-adic LWE as a candidate hard problem that would marry the hidden-radix philosophy with the cryptographic maturity of lattice-based hardness assumptions [3, Section 3.4 and 7.3].

   from 1.1 The Silent Radix and its Cryptographic Implications

Where to read

1. Abstract231 words · 1 min
2. 1. Introduction614 words · 3 min
3. 2. Mathematical Preliminaries354 words · 2 min
4. 3. The Hidden-Modulus LWE Problem637 words · 3 min
5. 4. Noise-Augmented SRE (N-SRE) Construction925 words · 4 min
6. 5. Security Analysis788 words · 3 min
7. 6. Parameter Selection and Performance284 words · 1 min
8. 7. Conclusion and Open Problems507 words · 2 min
9. Declarations127 words · 1 min